example of cohomology and Mayer-Vietoris sequence
Consider n-dimensional sphere .
Let and .
Of course both and are open (in ) and their union is . Furthermore, it can be easily seen, that their intersection can be contracted into ”big circle”, i.e. has homotopy type of . Also both and are contractible (they are homeomorphic to via stereographic projection). So, write part of a Meyer-Vietoris sequence (for the cohomology , where is a fixed Abelian group):
which shows that is isomorphic to for every and . So, in order to calculate cohomology groups of spheres, we only need to know the cohomology groups of . And those can be also calculated, if we once again apply previous schema. Note, that in the case of we have that has the homotopy type of a discrete space with two points. Therefore all their cohomology groups are trivial, except for (which can be easily calculated to be equal to , where * is a one-pointed space).
This schema can be used for other spaces like the torus (which can be also calculated from Kunneth’s formula).
|Title||example of cohomology and Mayer-Vietoris sequence|
|Date of creation||2013-03-22 19:13:27|
|Last modified on||2013-03-22 19:13:27|
|Last modified by||joking (16130)|